Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 210-211

Section Sum-frequency generation (SFG)

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: Sum-frequency generation (SFG)

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SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) are particular cases of three-wave SFG. We consider here the general situation where the two incident beams at ω1 and ω2, with [\omega_1\,\lt\,\omega_2], interact with the generated beam at ω3, with [\omega_3=\omega_1+\omega_2], as shown in Fig.[link]. The phase-matching configurations are given in Table[link].


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Frequency up-conversion process [\omega_1+\omega_2=\omega_3]. The beam at ω1 is mixed with the beam at ω2 in the nonlinear crystal NLC in order to generate a beam at ω3. [P^{\omega_1,\omega_2,\omega_3}] are the different powers.

From the general point of view, SFG is a frequency up-conversion parametric process which is used for the conversion of laser beams at low circular frequency: for example, conversion of infrared to visible radiation.

The resolution of system ([link] leads to Jacobian elliptic functions if the waves at ω1 and ω2 are both depleted. The calculation is simplified in two particular situations which are often encountered: on the one hand undepletion for the waves at ω1 and ω2, and on the other hand depletion of only one wave at ω1 or ω2. For the following, we consider plane waves which propagate in a direction without walk-off so we consider a single wave frame; the energy distribution is assumed to be flat, so the three beams have the same radius wo. SFG ([\omega_1+\omega_2=\omega_3]) with undepletion at [\omega_1] and [\omega_2]

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The resolution of system ([link] with [E_1(X,Y,0)\ne 0], [E_2(X,Y,0)\ne 0], [\partial E_1(X,Y,Z)/\partial Z=\partial E_2(X,Y,Z)/\partial Z=0] and [E_3(X,Y,0)= 0], followed by integration over [(X,Y)], leads to[\eqalignno{P^{\omega_1}(L)&=(T^{\omega_1})^2P^{\omega_1}(0)&(\cr P^{\omega_2}(L)&=(T^{\omega_2})^2P^{\omega_2}(0)&(\cr P^{\omega_3}(L)&=BP^{\omega_1}(0)P^{\omega_2}(0){L^2\over w_o^2}\sin c^2{\Delta k\cdot L\over 2}&(}%fd1.7.3.83]with[B_{\rm SFG}={72\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{\omega_3}T^{\omega_1}T^{\omega_2}\over n^{\omega_3}n^{\omega_1}n^{\omega_2}}\quad({\rm W}^{-1})]in the same units as equation ([link]. SFG ([\omega_s+\omega_p=\omega_i]) with undepletion at [\omega_p]

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[(\omega_s,\omega_p,\omega_i) = (\omega_1,\omega_2,\omega_3)] or [(\omega_2,\omega_1,\omega_3)].

The undepleted wave at ωp, the pump, is mixed in the nonlinear crystal with the depleted wave at ωs, the signal, in order to generate the idler wave at [\omega_i=\omega_s+\omega_p]. The integrations of the coupled amplitude equations over ([X,Y,Z]) with [E_s(X,Y,0)\ne 0], [E_p(X,Y,0)\ne 0], [\partial E_p(X,Y,Z)/\partial Z=0] and [E_i(X,Y,0)= 0] give[\eqalignno{P_p(L)&=T_p^2P_p(0)&(\cr P_i(L)&={\omega_i\over \omega_s}P_s(0)\Gamma^2L^2{\sin^2\{\Gamma^2L^2+[(\Delta k\cdot L)/2]^2\}^{1/2}\over \Gamma^2L^2+[(\Delta k\cdot L)/2]^2}&\cr&&(\cr P_s(L)&=P_s(0)\left[1-{\omega_s\over\omega_i}{P_i(L)\over P_s(0)}\right],&(}%fd1.7.3.86]with [\Delta k=k_i-(k_s+k_p)] and [\Gamma^2=[B_sP_p(0)]/w_o^2], where[B_s={8\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over \lambda_s\lambda_i}{T_sT_pT_i\over n_sn_pn_i}.]Thus, even if the up-conversion process is phase-matched ([\Delta k=0]), the power transfers are periodic: the photon transfer efficiency is then 100% for [\Gamma L=(2m+1)(\pi/2)], where m is an integer, which allows a maximum power gain [\omega_i/\omega_s] for the idler. A nonlinear crystal with length [L = (\pi/2\Gamma)] is sufficient for an optimized device.

For a small conversion efficiency, i.e. ΓL weak, ([link] and ([link] become[P_i(L)\simeq P_s(0){\omega_i\over \omega_s}\Gamma^2L^2\sin c^2{\Delta k\cdot L\over2}\eqno(]and [P_s(L)\simeq P_s(0).\eqno(]The expression for Pi(L) with [\Delta k=0] is then equivalent to ([link] with [\omega_p = \omega_1] or [\omega_2], [\omega_i=\omega_3] and [\omega_s = \omega_2] or [\omega_1].

For example, the frequency up-conversion interaction can be of great interest for the detection of a signal, ωs, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, Pi(L)/Ps(0), requires both wide spectral and angular acceptance bandwidths with respect to the signal. The double non-criticality in frequency and angle (DNPM) can then be used with one-beam non-critical non-collinear phase matching (OBNC) associated with vectorial group phase matching (VGPM) (Dolinchuk et al., 1994[link]): this corresponds to the equality of the absolute magnitudes and directions of the signal and idler group velocity vectors i.e. [{\rm d}\omega_i/{\rm d}{\bf k}_i={\rm d}\omega_s/{\rm d}{\bf k}_s].


Dolinchuk, S. G., Kornienko, N. E. & Zadorozhnii, V. I. (1994). Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion. Infrared Phys. Technol. 35(7), 881–895.

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